diferantial equation

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(x+y)dy/dx=x-y.

Let y=vx, dy/dx=v+xdv/dx.

Substitute for y:

(x+vx)dy/dx=x-vx.

Divide through by x and substitute for dy/dx:

(1+v)(v+xdv/dx)=1-v,

v+xdv/dx+v²+vxdv/dx=1-v,

v²+2v-1+x(v+1)dv/dx=0,

(v²+2v-1)/(v+1)=-xdv/dx.

The variables separate:

(v+1)dv/(v²+2v-1)=-dx/x.

Note that d(v²+2v-1)/dv=2(v+1), so integrate:

½ln(v²+2v-1)=ln|a/x|, where a is a constant.

ln(v²+2v-1)=2ln|a/x|=ln(a/x)².

Therefore, v²+2v-1=(a/x)².

But v=y/x, so:

y²/x²+2y/x-1=a²/x².

Multiply through by x²:

y²+2xy-x²=a². This is the equation of a “tilted” hyperbola:

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